Planck units

This article is about the system of units; for the scale see: Planck scale.

The Planck units form a system of natural units for the physical quantities.

They are calculated directly as products and quotients of the fundamental constants of nature from:

Gravitational constant $\textstyle G$
Speed of light $\textstyle c$
reduced Planckian quantum of action $\textstyle \hbar$
Boltzmann constant $\textstyle k_{{\mathrm {B}}}$
Coulomb constant $k_{\mathrm {C} }={\frac {1}{4\pi \textstyle \varepsilon _{0}}}$ (with the electric permittivity of the vacuum ε $\textstyle \varepsilon _{0}$ ).

Expressed in Planck units, these natural constants (or certain conventional multiples of them) therefore all have the numerical value 1. In this system of units, many calculations are then numerically simpler. Planck units are named after Max Planck, who remarked in 1899 that with his discovery of the quantum of action, enough fundamental constants of nature were now known to define universal units for length, time, mass, and temperature.

The importance of the Planck units lies, on the one hand, in the fact that the Planck units mark minimal limits (e.g. for length and time) up to which we can distinguish cause and effect. This means that beyond this limit the physical laws known so far are no longer applicable, e.g. in the theoretical elucidation of the processes shortly after the Big Bang (see Planck scale).

On the other hand, it is true, as Planck put it, that the Planck units "independent of special bodies or substances necessarily retain their meaning for all times and for all, also extraterrestrial and extra-human cultures and [...] can therefore be called 'natural units of measurement'", i.e., our laws of nature are universally applicable, understandable and communicable in the cosmos down to the Planck units.

Definitions

Basic sizes

The Planck units result from a simple dimension consideration. They result as mathematical expressions of the dimension of a length, time and mass, respectively, which contain only products and quotients of suitable powers of , and $\hbar$ . If one additionally uses the electric permittivity of the vacuum ε $\varepsilon _{0}$ and the Boltzmann constant $k_{\mathrm {B} }$ , then a Planck charge and a Planck temperature can also be determined as further basic quantities. The Planck charge satisfies the condition that the gravitational force between two Planck masses and the electromagnetic force between two Planck charges are equally strong: $m_{\mathrm {P} }^{2}G/\;\!l_{\mathrm {P} }^{2}=q_{\mathrm {P} }^{2}/4\pi \varepsilon _{0}\,l_{\mathrm {P} }^{2}$ .

Name	Size	Dimension	Term	Value in SI units	In other units
Planck length	Length	L	$l_\mathrm{P} = \sqrt{\frac{\hbar\,G}{c^3}}$	1,616 255(18) · 10−35 m	3,054 · 10−25 a0
Planck mass	Mass	M	$m_\mathrm{P} = \sqrt{\frac{\hbar\,c}{G}}$	2.176 434(24) - 10-8 kg	1.311 - ¹⁰¹⁹ u, 1.221 - ¹⁰¹⁹ GeV/c2
Planck time	Time	T	$\!\,t_{\mathrm {P} }={\sqrt {\frac {\hbar \,G}{c^{5}}}}={\frac {l_{\mathrm {P} }}{c}}$	5,391 247(60) · 10−44 s
Planck temperature	Temperature	Θ	$\!\,T_{\mathrm {P} }={\frac {m_{\mathrm {P} }\,c^{2}}{k_{\mathrm {B} }}}$	1,416 784(16) · 1032 K
Planck charge	Cargo	Q	$q_{\mathrm {P} }={\sqrt {4\pi \varepsilon _{0}\ \hbar \,c}}$	1,875 545 956(41) · 10−18 C	11,71 e

The formula symbols mean:

= speed of light
= gravitational constant
$\varepsilon _{0}$ = Electric field constant
$k_{\mathrm {B} }$ = Boltzmann constant
$\hbar$ = reduced Planckian quantum of action

Instead of $\,G$ sometimes $\,8\pi G$ set to one, then the mass unit is the reduced Planck mass:

${\overline {m_{\mathrm {P} }}}={\sqrt {\frac {\hbar c}{8\pi G}}}\approx 4{,}340\,\mu \mathrm {g}$ .

With the definition of a corresponding reduced Planck charge ${\overline {q_{\mathrm {P} }}}={\sqrt {\frac {\hbar \,c\,\varepsilon _{0}}{2}}}$ then the above equality of forces is preserved.

Derived quantities

In addition to these five basic quantities, the following derived quantities are also used:

Name	Size	Dimension	Term	Value in SI units
Planck area	Area	L2	$l_\mathrm{P}^2 = \frac{\hbar G}{c^3}$	2,612 · 10−70 ^m2
Planck volume	Volume	L3	$l_\mathrm{P}^3 = \sqrt{\frac{\hbar G}{c^3}}^{\,3}$	4,222 · 10−105 ^m3
Planck Energy	Energy	ML2T-2	$E_\mathrm{P} = m_\mathrm{P} c^2 = \frac{\hbar}{t_\mathrm{P}} = \sqrt{\frac{\hbar c^5}{G}}$	1.956 - ¹⁰⁹ J= 1.2209 - 1028 eV= 543.4 kWh
Planck pulse	Impulse	MLT-1	$m_\mathrm{P} c = \frac{\hbar}{l_\mathrm{P}} = \sqrt{\frac{\hbar c^3}{G}}$	6.525 kg m-s-1
Planck force	Force	MLT-2	$F_\mathrm{P} = \frac{E_\mathrm{P}}{l_\mathrm{P}} = \frac{\hbar}{l_\mathrm{P} t_\mathrm{P}} = \frac{c^4}{G}$	1,210 · 1044 N
Planck power	Power	ML2T-3	$P_\mathrm{P} = \frac{E_\mathrm{P}}{t_\mathrm{P}} = \frac{\hbar}{t_\mathrm{P}^2} = \frac{c^5}{G}$	3,628 · ¹⁰⁵² W
Planck density	Density	ML-3	$\rho_\mathrm{P} = \frac{m_\mathrm{P}}{l_\mathrm{P}^3} = \frac{\hbar t_\mathrm{P}}{l_\mathrm{P}^5} = \frac{c^5}{\hbar G^2}$	5.155 - 1096 kg-m-3
Planck angular frequency	Circular frequency	T−1	$\omega_\mathrm{P} = \frac{1}{t_\mathrm{P}} = \sqrt{\frac{c^5}{\hbar G}}$	1,855 · 1043 s−1
Planck pressure	Print	ML-1T-2	$p_\mathrm{P} = \frac{F_\mathrm{P}}{l_\mathrm{P}^2} = \frac{\hbar}{l_\mathrm{P}^3 t_\mathrm{P}} =\frac{c^7}{\hbar G^2}$	4.633 - ¹⁰¹¹³ Pa
Planck current	Electric current	QT-1	$I_\mathrm{P} = \frac{q_\mathrm{P}}{t_\mathrm{P}} = \sqrt{\frac{c^6 4 \pi \varepsilon_0}{G}}$	3,479 · 1025 A
Planck voltage	Electrical voltage	ML2T-2Q-1	$U_\mathrm{P} = \frac{E_\mathrm{P}}{q_\mathrm{P}} = \frac{\hbar}{t_\mathrm{P} q_\mathrm{P}} = \sqrt{\frac{c^4}{G 4 \pi \varepsilon_0} }$	1,043 · 1027 V
Planck impedance	Resistance	ML2T-1Q-2	$Z_{\mathrm {P} }={\frac {U_{\mathrm {P} }}{I_{\mathrm {P} }}}={\frac {\hbar }{q_{\mathrm {P} }^{2}}}={\frac {1}{4\pi \varepsilon _{0}c}}={\frac {Z_{0}}{4\pi }}$	29,98 Ω
Planck acceleration	Acceleration	LT-2	$g_\mathrm{P} = \frac{F_\mathrm{P}}{m_\mathrm{P}} = \sqrt{\frac{c^7}{\hbar G}}$	5.56 - ¹⁰⁵¹ m-s-2
Planck magnetic field	Magnetic flux density	MQ-1T-1	$B_{\mathrm {P} }={\sqrt {{\frac {\mu _{0}}{4\pi }}p_{\mathrm {P} }}}={\sqrt {\frac {c^{5}}{\hbar G^{2}4\pi \varepsilon _{0}}}}$	2,1526 · ¹⁰⁵³ T
Planck magnetic flux	Magnetic flux	ML2T-1Q-1	$\phi _{\mathrm {P} }={\frac {E_{\mathrm {P} }}{I_{\mathrm {P} }}}={\sqrt {\frac {\hbar }{4\pi \varepsilon _{0}c}}}$	5.6227 - 10-17 Wb

The Planck unit for the angular momentum results from the product of Planck length and Planck momentum to the value $\hbar$ . This is just the unit of angular momentum quantization known from quantum mechanics.

The Planck area $l_\mathrm{P}^2$ plays an important role, in particular, in string theories and in black hole entropy considerations related to the holographic principle.

History

At the end of the 19th century, during his investigations on the theory of radiation of black bodies, for which he received the Nobel Prize in Physics two decades later, Planck discovered the last natural constant necessary for the definition of Planck units, the quantum of action later named after him. He recognized the possibility of using it to define a universally valid system of units and mentioned it in a lecture "On Irreversible Radiation Processes." The following quotation gives an impression of the importance Planck attached to these units

"... the possibility is given to establish units [...] which, independent of special bodies or substances, necessarily retain their meaning for all times and for all, also extraterrestrial and extrahuman cultures, and which therefore can be called 'natural units of measurement'."

- Max Planck

Although Planck dedicated a chapter (§ 159. Natural units of measurement) of his book "Theory of Thermal Radiation" published in 1906 to this system of units and also took up this topic again later, it was not used even within physics. The disadvantages that the value of the gravitational constant was not (and still is) known exactly enough for the use in a system of units, and that practically relevant quantities - expressed in its units - would have absurd numerical values, were not opposed by any advantage, because in no physical theory the quantum of action and the gravitational constant appeared at the same time.

Only after first work on the unification of quantum theory and gravitation in the late 1930s, the later field of application of Planck units emerged. By the time John Archibald Wheeler and Oskar Klein published on the Planck length as the limit of applicability of general relativity in 1955, Planck's proposal had been all but forgotten. After the "rediscovery" of Planck's proposals for such a system of measurements, the name Planck units then became common from 1957.

However, the Planck units in use today differ from Planck's original units because, as quantum mechanics has developed, it has become apparent that $\hbar = \frac {h}{2\pi}$ the more practical natural unit than the chosen by Planck.

Present meaning

If equations containing the natural constants , and $\hbar$ in Planck units, the constants can be omitted. This greatly simplifies the equations in certain disciplines of theoretical physics, such as general relativity, quantum field theories, and the various approaches to quantum gravity.

Planck units also provide an alternative view of the fundamental forces of nature, whose strength is described in the International System of Units(SI) by very different coupling constants. Using the Planck units, the situation is as follows: Between two particles having exactly the Planck mass and the Planck charge, the gravitational force and the electromagnetic force would be exactly equal. The different strength of these forces in our world is the consequence of the fact that a proton and an electron, respectively, have a charge of about 0.085 Planck charges, while their masses are smaller than the Planck mass by 19 and 22 orders of magnitude, respectively. So the question: "Why is gravity so weak?" is equivalent to the question: "Why do the elementary particles have such small masses?

Various physicists and cosmologists deal with the question, whether we could notice, if dimensional physical constants would change slightly, and how the world would look like in case of larger changes. Such speculations have been made, among others, about the speed of light and the gravitational constant , the latter already since about 1900 in the expansion theory of the earth. The atomic physicist George Gamow means in his popular scientific book Mr. Tompkins in the wonderland that a change of would result in clear changes.

Questions and Answers

Q: What are Planck units?

A: Planck units are physical units of measurement first developed by Max Planck, based on four physical constants found in nature. When used to express any of these four physical constants, the value is 1.

Q: What are the four basic Planck units based on?

A: The four basic Planck units are based only on four physical constants found in nature, which include the speed of light in a vacuum (c), the gravitational constant (G), the reduced Planck constant (ħ) and the Boltzmann constant (kB).

Q: Why are they called natural units?

A: They are called natural units because they come only from properties of nature and not from any human construct.

Q: How do natural units help physicists?

A: Natural units help physicists to simplify several recurring algebraic expressions of physical law and reframe questions. They also eliminate human centered arbitrariness from the system of units.

Q: What theories do each of these constants have at least one fundamental physical theory associated with them?

A: c has special relativity associated with it, G has general relativity and Newton's law of universal gravitation associated with it, ħ has quantum mechanics associated with it, ε0 has electrostatics associated with it, and kB has statistical mechanics and thermodynamics associated with it.

Q: Why may Planck Units sometimes be semi-humorously referred to as "God's Units"?

A: They may be referred to as "God's Units" because they eliminate human centered arbitrariness from the system of units and some physicists argue that communication with extraterrestrial intelligence would have to use such a system of units to make common reference to scale.